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Pro-P Groups Acting on Trees with Finitely Many Maximal Vertex

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Pro-P Groups Acting on Trees with Finitely Many Maximal Vertex Pro-p groups acting on trees with finitely many maximal vertex stabilizers up to conjugation Zo´eChatzidakis and Pavel Zalesskii ∗ November 18, 2020 Abstract We prove that a finitely generated pro-p group G acting on a pro-p tree T splits as a free amalgamated pro-p product or a pro-p HNN-extension over an edge stabilizer. If G acts with finitely many vertex stabilizers up to conjugation we show that it is the fundamental pro-p group of a finite graph of pro-p groups (G; Γ) with edge and vertex groups being stabilizers of certain vertices and edges of T respectively. If edge stabilizers are procyclic, we give a bound on Γ in terms of the minimal number of generators of G. We also give a criterion for a pro-p group G to be accessible in terms of the first 1 cohomology H (G; Fp[[G]]). 1 Introduction The dramatic advance of classical combinatorial group theory happened in the 1970's, when Bass-Serre theory of groups acting on trees changed completely the face of the theory. The profinite version of Bass-Serre theory was developed by Luis Ribes, Oleg Melnikov and the second author because of the absence of the classical methods of combinatorial group theory for profinite groups. However it does not work in full strength even in the pro-p case. The reason is that if a pro-p group G acts on a pro-p tree T then a maximal subtree of the quotient graph GnT does not always exist and even if it exists it does not always lift to T . As a consequence the pro-p version of Bass-Serre theory does not give subgroup structure theorems the way it does in the classical Bass-Serre theory. In fact, for infinitely generated pro-p subgroups there are counter examples. The objective of this paper is to study the situation when G has only finitely many vertex stabilizers up to conjugation and in this case we can prove the main Bass-Serre theory structure theorem. Theorem 5.1. Let G be a finitely generated pro-p group acting on a pro-p tree T with finitely many maximal vertex stabilisers up to conjugation. Then G is the fundamental group of a ∗The second author partially supported by CNPq and FAPDF. 1 reduced finite graph of finitely generated pro-p groups (G; Γ), where each vertex group G(v) and each edge group G(e) is a maximal vertex stabilizer Gv~ and an edge stabilizer Ge~ respectively (for some v;~ e~ 2 T ). In the abstract situation, a finitely generated (abstract) group G acting on a tree has a G-invariant subtree D such that GnD is finite and so has automatically finitely many maximal vertex stabilizers up to conjugation. In the pro-p situation such an invariant subtree does not exist in general and the existence of it in the case of only finitely many stabilizers up to conjugation is not clear even if vertex stabilizers are finite. Nevertheless, for a finitely generated pro-p group acting on a pro-p tree we can prove a splitting theorem into an amalgamated product or an HNN-extension. Theorem 4.2. Let G be a finitely generated pro-p group acting on a pro-p tree T without global fixed points. Then G splits non-trivially as a free amalgamated pro-p product or pro-p HNN-extension over some stabiliser of an edge of T . This in turn allows us to prove that a non virtually cyclic pro-p group acting on a pro-p tree with finite edge stabilizers has more than one end. Theorem 4.5. Let G be a finitely generated pro-p group acting on a pro-p tree with finite edge 1 ∼ stabilizers and without global fixed points. Then either G is virtually cyclic and H (G; Fp[[G]]) = 1 Fp (i.e. G has two ends) or H (G; Fp[[G]]) is infinite (i.e. G has infinitely many ends). Theorem 4.2 raises naturally the question of accessibility; namely whether we can continue to split G into an amalgamated free product or HNN-extension forever, or do we reach the situation after finitely many steps where we can not split it anymore. The importance of this is underlined also by the following observation: if a pro-p group G acting on a pro-p tree T is accessible with respect to splitting over edge stabilizers, then by Theorem 4.2 this implies finiteness of the maximal vertex stabilizers up to conjugation and so Theorem 5.1 provides the structure theorem for G. In the abstract situation, accessibility was studied by Dunwoody ([3], [4]) for splitting over finite groups, and by Bestvina and Feighn ([1]) over an arbitrary family of groups. In the pro-p case accessibility was studied by Wilkes ([24]) where a finitely generated not accessible pro-p group was constructed. For a finitely generated pro-p group acting faithfully and irreducibly on a pro-p tree (see Section 2 for definitions) no such example is known. The next theorem gives a sufficient condition of accessibility for a pro-p group; we do not know whether the converse also holds (it holds in the abstract case). 1 Theorem 6.10. Let G be a finitely generated pro-p group. If H (G; Fp[[G]]) is a finitely gen- 1 erated Fp[[G]]-module, then G is accessible. We show here that finitely generated pro-p groups are accessible with respect to cyclic subgroups and in fact give precise bounds. 1proved by G. Wilkes independently in [25]. 2 Theorem 6.6. Let G = π1(Γ; G) be the fundamental group of a finite graph of pro-p-groups, with procyclic edge groups, and assume that d(G) = d ≥ 2. Then (assuming that the graph is reduced), the vertex groups are finitely generated, the number of vertices of Γ is ≤ 2d − 1, and the number of edges of Γ is ≤ 3d − 2. Observe that Theorem 6.6 contrasts with the abstract groups situation where for finitely generated groups the result does not hold (see [5]). As a corollary we deduce the bound for pro-p limit groups (pro-p analogs of limit groups introduced in [10], see Section 6 for a precise definition). Corollary 6.7. Let G be a pro-p limit group. Then G is the fundamental group of a finite graph of finitely generated pro-p groups (G; Γ), where each edge group G(e) is infinite procyclic. Moreover, jV (Γ)j ≤ 2d − 1, and jE(Γ)j ≤ 3d − 2, where d is the minimal number of generators of G. It is worth mentioning that for abstract limit groups the best known estimate for jV (Γ)j is 1 + 4(d(G) − 1), proved by Richard Weidmann in [22, Theorem 1]. In Section 7 we investigate Howson's property for free products with procyclic amalgamation and HNN-extensions with procyclic associated subgroups. In Section 8.1 we apply the results of Section 7 to normalizers of procyclic subgroups. Theorem 8.3. Let C be a procyclic pro-p group and G = G1 qC G2 be a free amalgamated pro-p product or a pro-p HNN-extension G = HNN(G1; C; t) of Howson groups. Let U be a procyclic g g subgroup of G and N = NG(U). Assume that NGi (U ) is finitely generated whenever U ≤ Gi. If K ≤ G is finitely generated, then so is K \ N. Section 2 contains basic notions and facts of the theory of pro-p groups acting on trees used in the paper. The following sections are devoted to the proofs of the results mentioned above. Aknowledgement. This research was started while the second author was visiting the ENS as an invited professor, and the authors thank the ENS for this hospitality. 2 Notation, definitions and basic results 2.1. Notation. If a pro-p group G continuously acts on a profinite space X we call X a G- space. H1(G) denotes the first homology H1(G; Fp) and is canonically isomorphic to G=Φ(G). −1 g −1 If x 2 T and g 2 G, then Ggx = gGxg . We shall use the notation h = g hg for conjugation. If H a subgroup of G, HG will stand for the (topological) normal closure of H in G. If G is an abstract group Gb will mean the pro-p completion of G. 2.2. Conventions. Throughout the paper, unless otherwise stated, groups are pro-p, sub- groups will be closed and morphisms will be continuous. Finite graphs of groups will be proper and reduced (see Definitions 2.12 and 2.15). Actions of a pro-p group G on a profinite graph 3 Γ will a priori be supposed to be faithful (i.e., the action has no kernel), unless we consider actions on subgraphs of Γ. Next we collect basic definitions, following [17]. 2.1 Profinite graphs Definition 2.3. A profinite graph is a triple (Γ; d0; d1), where Γ is a profinite (i.e. boolean) space and d0; d1 :Γ ! Γ are continuous maps such that didj = dj for i; j 2 f0; 1g. The elements of V (Γ) := d0(G) [ d1(G) are called the vertices of Γ and the elements of E(Γ) := Γ n V (Γ) are called the edges of Γ. If e 2 E(Γ), then d0(e) and d1(e) are called the initial and terminal vertices of e. A vertex with only one incident edge is called pending.
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